How do I create a precise, tight fitting render border around an object?
Full disclosure, the add-on I'm about to recommend is a paid for add-on created by me.

'Animated Render Border', my add-on, allows the user to select an object from a drop-down list and automatically detect where the render border should be positioned so the object is encompassed by it:



You can choose whether it does this via the object's bounding box, or more precisely, using its vertices (if it is a mesh, curve, armature (uses bones) or lattice). It also has a mode to allow a more manual fine tuning ('Keyframe' mode).

As the name might suggest, this is actually intended for objects that are animated that you want to be contained within the render border, but it works perfectly fine for stills.

There are some limits in that modifiers that alter the object's geometry aren't currently supported and there are some objects which can't be 'tracked', like empties, lamps and cameras.